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Nos es muy grato presentarles
a los estudiantes estudiantes y profesional profesionales es de topografía topografía el presente presente texto, que es el resultado de un gran trabajo de consulta y compilación bibliográfica de la información dispersa en distintos medios como libros, manuales e Internet, sobre el área de planimetría. En este libro recogimos la experiencia como estudiantes del programa de topografía de la Universidad del Quindío y el conocimiento de un gran profesional como lo es el topógrafo y fotogrametrista GILBERTO GÓMEZ GÓMEZ; que nos permite tener una visión de las necesidades del estudiante de dicho programa y de otros de la misma institución. Para hacer que este sea amplio en conceptos, criterios y conocimientos de las técnicas propias de los procesos planimetritos. De esta forma la finalidad del documento es servir de facilitador, apoyo y herramienta de mejoramiento de las dinámicas de enseñanza y aprendizaje de dicha materia. Ya que pensamos que el desarrollo lógico y claro de la sustentación teórica, con ejemplos prácticos y una descripción descripción apropiada apropiada de las técnicas técnicas de trabajo, trabajo, lograra lograra aclarar y afianzar afianzar las nociones básicas adquiridas, haciéndolo una alternativa de consulta muy interesante para el estudiante. Así pues en el texto se pueden encontrar todos los temas necesarios para la comprensión de la planimetría; divididos es 8 capítulos. El capítulo 1(Preámbulo) en el que están aquellos conocimientos de matemáticas esenciales para entender las ideas relacionadas con la planimetría. El capítulo 2 (Medidas y Errores) contiene la explicación de la teoría de errores a través de una perspectiva sencilla, apropiada al nivel del estudiante. El capítulo 3 (Teodolito) se encuentran las características de composición, funcionalidad y las nociones de manejo de los tránsitos. Capítulo 4 (Ángulos) allí se tienen la tipologia de ángulos, sus relaciones y la manera de medirlos. Capítulo 5 (Distancias) en el encontramos la clasificación de las distancias y los diversos sistemas de medición empleados en los trabajos planimetricos. El capítulo 6 (Poligonales) allí se reúnen los conocimientos de geometría para explicar la clasificación de las poligonales topográficas; en este también encontramos la explicación teórica de los procedimientos y criterios del levantamiento de poligonales con transito. Capítulo 7 (Levantamientos) en el están las descripciones de los procesos técnicos de trabajo en levantamientos planimétricos y la forma lógica de llevar a cabo cada trabajo para obtener los mejores resultados. Capítulo 8 (Cálculos) están las explicaciones imprescindibles de los cálculos que permiten llevar la información obtenida en campo a una serie de datos que pueden ser utilizados en la localización de proyectos
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1
1
1.1.2 El Punto
La Línea 1.1.4 Recta – Segmento 1.1.5 Ángulos 1.1.6 Triángulos 1.1.7 Cuadriláteros 1.1.8 Polígonos 1.1.9 Circunferencias Circunferencias 1.1.10 Áreas 1.1.3
2.1 Comparaciones cualitativas 2.2 Comparaciones cuantitativas 2.3 Concepto de Medida 2.3.1 Propiedades de una Medida 2.3.2 Formas de Medir 2.3.2.1 Contar 2.3.2.2 Medición directa 2.3.2.3 Medición indirecta 2.4 Distribución Normal
1.2.1 Función de primer grado 1.2.2 Función cuadrática – resolución de ecuaciones 1.2.3 Sistemas de ecuaciones
16
1.3.1 Coordenadas polares 1.3.2 Coordenadas rectangulares o cartesianas 1.3.2.1 Distancia entre 2 puntos 1.3.3 Línea recta 1.3.3.1 Posición relativa entre dos rectas
17
2.5 Precisión y Exactitud 2.5.1 Precisión 2.5.2 Exactitud 2.6 Incertidumbre de las medidas 2.7 Tipos de Errores: 2.7.1 Sistemáticos Sistemáticos 2.7.2 Aleatorios 2.8 Causas de Errores: Naturales Instrumentales Personales 2.9 Error Real 2.10 Error Relativo
37
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Disposición de los ejes del tránsito 3.2 Base nivelante 3.3 Nivel tubular 3.4 Sensibilidad del nivel tubular 3.1
3.5 Alidada o montante 3.6 Limbos 3.6.1 Vernier o nonio 3.7 Sistemas de fijación o movimiento lento 3.8 Sistemas de compensación vertical 3.9 Sistemas de lecturas de limbos 3.9.1 Lectura simple 3.9.2 Lectura por coincidencia 3.9.3 Microscopio a escala 3.9.4 Microscopio con micrómetros micrómetros 3.10 Telescopio 3.10.1 Ocular 3.10.2 Retícula 3.10.3 Sistema de enfoque 3.10.4 Objetivo 3.11 Controles y errores del tránsito 3.11.1 Error de inclinación del eje vertical 3.11.2 Error de falta de perpendicularidad entre el eje vertical y el eje basculante 3.11.3 Error de colimación 3.11.4 Error de puntería 3.11.5 Error de graduación
55
4.1.3 Ángulo horario 4.1.4 Ángulo contrahorario 4.1.5 Deflexiones 4.2 Relaciones entre ángulos 4.2.1 Azimut - contrazimut 4.2.2 Azimut – rumbo 4.2.3 Rumbo – azimut 4.2.4 Azimut – deflexiones 4.2.5 Deflexiones – azimut 4.2.6 Ángulo horario – deflexiones 4.2.7 Ángulo horario – deflexiones – azimut 4.3 Ángulos verticales 4.4 Medición de ángulos 4.4.1 Medición utilizando el tránsito 4.4.1.1 Medida sencilla 4.4.1.2 Método de repeticiones 4.4.1.3 Método de reiteraciones
5.1 Métodos medición de Distancias Horizontales 5.1.1 Medición a pasos 5.1.2 Medición con odómetro 5.1.3 Medición con cinta Alineación Verticalidad Ubicación de puntos Anotaciones Cintas usadas actualmente 5.1.4Mediciones Horizontales en terreno inclinado
103 109
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137
6.1 Clasific ación
6.1.1 Poligonal abierta 6.1.2 Poligonal cerrada 6.1.3 Poligonal establecida por radiaciones desde una estación 6.2 Poligonales con tránsito 6.2.1 Ángulo horario o contrahorario 6.2.2 Método del azimut 6.2.3 Deflexiones
7.1 Levantamientos con cinta 7.1.1 Levantamientos de poligonales con cinta 7.2 Levantamientos con brújula y cinta 7.2.1 Levantamientos de detalles con brújula 7.3 Levantamientos con tránsito 7.4 Cronología del trabajo de una poligonal 7.5 Trabajo en Campo 7.5.1 Selección del método 7.5.2 Anotaciones de campo 7.5.3 Selección de puntos de estación 7.5.4 Estacionando el tránsito 7.5.5 Orientación
8.2 Cálculos de comprobación 8.2.1 Cierre angular 8.2.2 Error de cierre 8.2.3 Precisión en poligonales
8.3 Métodos de ajuste 8.3.1 Método de la brújula 8.3.2 Método del tránsito 8.3.3 Método XY 8.3.4 Método de Crandall 142
8.4 Cálculo de coordenadas
145
8.5 Cálculo de áreas 8.5.1 Métodos gráficos Planímetro de puntos Planímetro polar Papel milimetrado milimetrado Áreas por descomposición en triángulos 8.5.2 Métodos analíticos Método de gauss Método de coordenadas polares Método de los trapecios Método de Simpson 8.6 Omisión de datos de una poligonal 8.6.1 Longitud y rumbo de un lado. 8.6.2 Longitud de un lado y rumbo de otro. 8.6.3 Longitud de dos lados. 8.6.4 Rumbo de dos lados.
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En este capítulo
se hará un repaso de conceptos y elementos de álgebra, geometría, geometría analítica, trigonometría y escalas; para hacer más sencillo el entendimiento de los conceptos topográficos que se van a estudiar en el transcurso del libro.
Veamos ahora algunos axiomas, corolarios y fórmulas necesarios en los cálculos y ejercicios topográficos.
El vocablo geometría, tiene sus orígenes en las raíces griegas, que que significa tierra y que que viene de que que significa medida, por lo tanto podemos inferir que la geometría es la ciencia de medir la tierra. Se dice también que la geometría es la parte de las matemáticas que trata de las propiedades y medidas de la extensión, estudiando la forma y dimensiones de las figuras geométricas.
es la medida del espacio ocupada por un cuerpo. La es Una figura geométrica es un espacio cerrado c errado por puntos, líneas o superficies.
Ahora veamos unos axiomas y corolarios muy importantes en geometría sobre líneas, ángulos, triángulos, polígonos y circunferencia.
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Planimetría________________________________________________________________
También podemos clasificar las líneas de acuerdo con su posición:
Línea horizontal
Línea vertical
Línea oblicua
La recta es “ilimitada”, se considera prolongada indefinidamente. indefinidamente. Dos rectas solo se cortan en un solo punto o son son coincidentes. Dos rectas son paralelas paralelas si a pesar pesar de prolongarlas indefinidamente nunca nunca se llegan a cortar. Si dos rectas se cortan, sus perpendiculares también lo harán. La línea recta es la distancia más corta entre dos puntos. Una semirecta o rayo es una porción de recta que se forma cuando sobre una recta se traza un punto que la divide en dos semirectas.
Se llama segmento al conjunto de puntos comprendidos entre dos puntos de una recta.
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Capitulo 1: Preámbulo
Un ángulo es la intersección de dos rayos o semirrectas. Los rayos se llaman lados del ángulo y el punto de unión recibe el nombre de vértice. Un ángulo se obtiene por la rotación de un rayo alrededor de su origen y lo podemos identificar por tres letras donde la del centro corresponde al vértice y las otras a puntos cualquiera de las semirrectas, también podemos identificarlos utilizando letras del alfabeto griego en su interior .
Existen cuatro elementos básicos que determinan un ángulo, estas son:
Lado origen El sentido de giro
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Planimetría________________________________________________________________
Correspondientes: uno interno y otro externo, no adyacentes al mismo lado de la secante
Dos rectas que se cortan forma cuatro ángulos iguales, dos a dos que se llaman opuesto por el vértice.
Opuestos por el vértice: Correspondientes: Alternos internos: Alternos externos:
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Capitulo 1: Preámbulo a) Lados: Son las líneas que lo limitan.
b) Vértices: son el lugar donde se unen los lados. Los triángulos los podemos clasificar según sus ángulos o sus lados
Equilátero: Es aquel que tiene sus tres lados iguales.
Isósceles: Es aquel que posee dos lados iguales.
Escaleno: Es aquel en el que todos sus lados son desiguales.
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Planimetría________________________________________________________________
Si en un triángulo rectángulo uno de sus ángulos agudos y su cateto adyacente son congruentes a los del otro triangulo rectángulo estos son congruentes congruentes entre sí. (ALA) Dos triángulos son congruentes si los lados de uno son iguales a los lados del otro triángulo. (LLL) Dos triángulos son congruentes si dos de sus ángulos y un lado de uno, son respectivamente congruentes a dos ángulos y un lado incluido del otro.(AAL) En un triangulo isósceles los ángulos de la base son iguales, siendo la base el lado de magnitud diferente. :
Es la perpendicular que pasa por el punto punto medio de un lado en un triángulo. El punto de intersección de las mediatrices recibe el nombre de circuncentro.
: Es el segmento trazado trazado desde un vértice hasta hasta el punto
medio del lado opuesto. El punto de intersección de las medianas de un triángulo recibe el nombre de baricentro.
El corta las medianas en un punto situado a 2/3 de su longitud a partir del vértice .
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Capitulo 1: Preámbulo acutángulo, el ortocentro es interior. interior. un punto llamado . Si el triángulo es acutángulo, En un triángulo rectángulo, cada cateto puede ser considerado como base y como altura. El ortocentro es, por tanto, el vértice del ángulo recto. Si el triángulo es obtusángulo el ortocentro se obtiene, prolongando las alturas, fuera del triángulo. Figura 9.
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Planimetría________________________________________________________________
Dos triángulos son semejantes cuando tienen sus lados homólogos proporcionales.
Figura de cuatro lados; Son cuadriláteros: los paralelogramos, cuadrados, rombos, rectángulos y trapecios.
Un paralelogramo es un cuadrilátero que tiene sus lados opuestos iguales y paralelos, como lo son el cuadrado, el rombo, y el rectángulo. Las diagonales de un paralelogramo se bisecan y dividen el paralelogramo en dos triángulos congruentes, igual sucede en el rombo, el cuadrado y el rectángulo. En el cuadrado todos los ángulos y los lados son iguales. En el rectángulo los ángulos son iguales y su medida es de 90°, los lados opuestos opuestos son iguales.
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Capitulo 1: Preámbulo
En el trapecio la altura es la separación entre las bases.
La paralela a las bases de un trapecio trazada por el punto medio de uno de los lados no paralelos es igual a la semisuma semisuma de las bases.(base media)
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Planimetría________________________________________________________________
La suma de los ángulos de de un polígono es de 360°.
es en el que dos o más de sus lados se cursan.
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Capitulo 1: Preámbulo circunferencia es la parte de ésta delimitada por dos puntos. Angulo en el centro es aquel cuyo vértice es el centro de circunferencia y cuyos lados son dos radios. El centro de la circunferencia es centro de simetría, y cualquier diámetro es eje de simetría. Una semicircunferencia es cada una de las partes determinadas por el diámetro. Un sector circular es una porción de círculo determinada por dos radios y un arco. El semicírculo es una porción de círculo limitada por la semicircunferencia. La corona circular es figura plana comprendida entre dos círculos concéntricos.
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Planimetría________________________________________________________________
: si tienen dos puntos puntos comunes.
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Capitulo 1: Preámbulo Los triángulos, aunque no sean regulares, tienen siempre circunferencia inscrita (tangente a sus tres
lados) y circunscrita (que pasa por sus tres vértices).
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Planimetría________________________________________________________________
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Capitulo 1: Preámbulo
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Planimetría________________________________________________________________
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En los trabajos de levantamientos, son esenciales esenciales las mediciones de distancias y/o ángulos y con este fin son empleados diferentes diferentes instrumentos, instrumentos, entre los que que se encuentra el teodolito. Este instrumento fundamentalmente se emplea para la medición de ángulos, por esto toma el nombre genérico de (goniómetro) (gonia ángulo y metrón medida). medida). Los goniómetros que se utilizan en topografía; son de plano horizontal o de plano vertical y los que pueden trabajar trabajar en ambos planos se denominan goniómetros universales, universales, este instrumento lo conocemos conocemos como teodolito o tránsito.
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Planimetría_____________________________________________________________ El inventor del termino teodolito teodolito “fue Leonard Leonard Dignes” Dignes” quien escribi escribió ó de la descripción descripción del instrumento en un texto con el nombre de The Construction of an Instrument Topographical Surveing Most Commodioly for all Manner of Measurations. Dicha descripción fue publicada en el siglo XVI por su hijo Thomas; a partir de este momento los topógrafos ingleses dedicaron sus esfuerzos al desarrollo de dicho instrumento, los teodolitos que resultaron de éste esfuerzo eran muy grandes con círculos horizontales de 3 ft de diámetro. El tamaño de estos limbos se debe a la relación que existe entre el tamaño del círculo y la precisión de medida angular ya que al ser más grandes permiten que se puedan marcar más divisiones en él. Como no se había desarrollado hasta el momento sistemas ópticos de lectura, era necesario utilizar círculos muy grandes para obtener una buena precisión, otra característica de los primeros teodolitos es que no poseían limbo vertical por lo tanto no se podían efectuar mediciones angulares verticales. Los teodolitos continuaron evolucionando y se construyeron equipos mas pequeños que conservaban la precisión de medida angular angular y en algunos se mejoraba, adicionalmente adicionalmente se les logro colocar círculos verticales, estos equipos equipos se les denomino denomino tránsitos. En la actualidad encontramos encontramos equipos con sistemas de lectura completamente electrónicos y con otros aditamentos que hacen mas fácil y aligeran el trabajo del topógrafo, claro que estos instrumentos siguen conservando sus finalidades iniciales.
Medir y trazar ángulos verticales Medir y trazar ángulos horizontales Trazar alineamientos Medición óptica de distancias Medición electrónica de distancias.
En los tránsitos para garantizar la medición exacta de los ángulos verticales tanto como horizontales y la medición óptica de distancias, éste debe cumplir unas condiciones en la disposición de los ejes, estas son: 1. el eje eje vertical vertical (vv) debe ser perpendicul perpendicular ar a el el eje eje del nivel (LL) 2. el eje eje vertica verticall (vv) (vv) debe debe ser ser perpend perpendicul icular ar al eje horizontal o basculante (HH) 3. el eje eje horizon horizontal tal (HH) (HH) debe debe ser ser perpendic perpendicular ular al eje eje de puntería (zz) Debido al el trabajo continuo con los instrumentos, estas disposiciones pueden cambiar, por lo cual se debe estar realizando un control del estado del instrumento. Cuando en esta revisión revisión se encuentren diferencias suficientemente suficientemente grandes para que reduzcan la calidad de las observaciones se deben llevar acabo las correcciones pertinentes.
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Capitulo 3: Teodolito
Es la parte del transito que descansa, sobre el plato del del trípode, provista de una placa de acoplamiento, dicha placa puede tener dos sistemas de fijación al trípode, uno de estos es el sistema de tornillo ( o el sistema de bayoneta ( También tiene unos tornillos de nivelación montados sobre la placa y un nivel circular o de blanco. Todo este conjunto puede ser calado en posición horizontal con ayuda de los tornillos de nivelación y un nivel circular (calado aproximado) o con un nivel tubular que se encuentra en la alidada. Al mover los tornillos podemos cambiar la inclinación de la base hasta lograr colocar la burbuja del nivel quede dentro de sus reparos centrales y así asegurar que la base del transito se encuentra en posición horizontal. Existen dos sistemas de tornillos para la base nivelante:
Sistema de tornillos verticales Sistema de tornillos horizontales
En el sistema de tornillos verticales; estos están dispuestos en un numero de tres tornillos para equipos europeos y cuatro tornillos en equipos de construcción americana que se encuentra colocados o montados en una cruceta, donde los extremos redondeados del tronillo operan en unos bujes fijos a la placa de acoplamiento, en este sistema el tornillo es fijo y lo que se mueve es la parte superior de la base modificando la altura instrumental. El sistema de tornillos horizontales es completamente distinto, en este los tornillo utilizan unas piezas giratorias cuyo centro de rotación es distinto al centro de la figura esto hace que se genere un pequeño movimiento vertical de la base, teniendo así la propiedad de que este sistema no modifica la altura instrumental y se elimina cualquier posibilidad de juego entre las piezas del sistema que si se presenta en el sistema de tornillos verticales. (
En la base nivelante se hacen presentes unos dispositivos para el centrado como el gancho en el tornillo de fijación al trípode, para la poner la plomada, plomada óptica, que esta compuesto de un
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Planimetría_____________________________________________________________
Como ya vimos el nivel tubular estos nos ayuda a nivelar el transito, consiste en un tuvo de cristal con unas divisiones en su parte superior que se encuentran uniformemente espaciadas llamadas reparos; este cilindro contiene su parte interior una fracción de arco de toro, así pues que si hiciéramos un corte longitudinal a un nivel en su parte interior se vería un arco de circulo .
El arco de toro se encuentra casi lleno de éter sulfúrico o alcohol y el espacio restante se encuentra lleno de aire formándose así una burbuja que siempre ocupa la parte más alta de dicho arco. Utilizando esta propiedad es que podemos asegurar la nivelación del instrumento, ( en la parte ( de dicha grafica se, encuentra un nivel montado montado en una base que está inclinada inclinada un ángulo alfa a la izquierda, por su propiedad la burbuja del nivel se desplaza a la derecha buscando la parte mas alta del mismo, en la parte de la grafica vemos la misma base pero con un ángulo de inclinación alfa igual a cero, por lo tanto la burbuja se encuentra en el centro del arco que en este caso es la parte mas alta de él, por tal motivo si se pasase un recta tangente por el punto medio de la curva interior del nivel, esta línea será horizontal con respecto a la vertical (eje del nivel). Para esta clase de niveles existen dos sistemas de apreciación, el nivel tubular de división aparente y el nivel de coincidencia este es un sistema muy ingenioso de disposición de prismas que nos permite yuxtaponer la imagen de la mitad izquierda y la derecha del nivel; cuando estas dos imágenes coinciden la base queda completamente nivelada (calada en la horizontal).
Esta dada por el radio de curvatura del arco de toro, miremos la ( , en este este encontram encontramos os dos niveles con radios de curvatura diferente que se encuentra inclinados en un mismo ángulo alfa, nivel
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Capitulo 3: Teodolito
La precisión del centrado centrado en un nivel de de división aparente aparente esta dada por 1/5 del intervalo intervalo de la separación entre los reparos (2mm) por tanto la precisión de este es de 0.4 mm en el caso de nivel de coincidencia el centrado es de 1/40 del intervalo de separación, esto quiere decir que tiene una precisión de 0.05mm aproximadamente.
Esta parte del instrumento esta formada por la base superior y el montante; en la base se encuentran el circulo horizontal graduado (limbo), los sistemas de fijación y movimiento lento de dicho limbo. El montante es el encargado de sustentar el eje horizontal o de alturas, por medio de dos soportes verticales, dentro de estos soporte verticales encontramos también el circulo graduado graduado vertical, los sistemas de fijación y de moviendo moviendo vertical de éste, las disposiciones disposiciones de prismas para la lecturas de los limbos, el el microscopio de lectura y los dispositivos dispositivos de compensación de colimación vertical. vertical. La alidada gira alrededor de de su eje vertical, por tal razón razón cuando el instrumento instrumento se encuentra en posición horizontal (nivelado), el telescopio montado sobre éste generan un plano perpendicular al eje vertical del aparato. Ya se hablo que dentro de todo este conjunto existen unos círculos graduados a los que llamamos llamamos limbos, veamos ahora que son estos dispositivos y en que consiste cada una de las partes nombradas.
Son unas escalas circulares que permiten la medición directa directa de grados y de múltiplos de 5, 10, 15, 20, 30 minutos; en sus comienzos dicho dispositivos dispositivos eran hecho de metal , en la actualida actualidad d son de cristal cristal y se puede puede leer por medio de unos sistemas ópticos de los cuales hablaremos mas
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Planimetría_____________________________________________________________ Los limbos en el transito, comprenden dos placas que se encuentran concéntricas y superpuestas, una de ellas esta fija y las otra móvil móvil junto con todo el conjunto conjunto (base –montante). –montante). En los tránsitos de tipo repetidor existe un sistema de pinzas que permite fijar el limbo horizontal, en una posición determinada, a la placa móvil móvil y que que éste gire también. también. Esto se emplea cuando se hace necesario visar a un punto punto con un ángulo determinado; la otra posibilidad de llevar a cabo dicha operación es visar a el punto dejar fijo el instrumento y mover el limbo haciendo haciendo uso de un tornillo que que nos permite este movimiento, a estos equipos los llamamos reiteradores; en este sistema lo que hace el tornillo es servir de piñón y hacer que la placa donde se encuentra montado el limbo horizontal se mueva mueva como lo vemos vemos en la
Como ya lo dijimos los limbos no permite leer los grados y unos múltiplos de minutos, existen algunos trabajos donde se requiere mas mas precisión angular, para esto se emplea un elemento que es denominado nonio o vernier.
La agudeza visual del ojo del del humano no le permite permite la apreciación directa de divisiones menores menores a la menor división de la regla o limbo que este utilizando. Por esto el matemático y astrónomo portugués Pedro Nunes, se ingenio un dispositivo dispositivo para precisar los valores de dichas apreciaciones; éste dispositivo fue perfeccionado por el geógrafo francés Pierre vernier de allí el nombre que este toma. Para las mediciones de tipo angular se emplean unos arcos secundarios que se encuentran en
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Capitulo 3: Teodolito La apreciación del nonio viene dada por:
=
Donde es la división mas pequeña de él limbo y es el numero de Divisiones del nonio
Si queremos leer un vernier debemos tener en cuenta:
• •
El cero del nonio indica las cantidades enteras del limbo La graduación del nonio que coincide con una de la escala principal indica la fracción adicional
Vamos a construir un sistema de lectura que nos permita, con el nonio, leer a 10” en un limbo de 10’. Procedimiento: Construimos un segmento de limbo con la graduación indicada de 10’ utilizando la formula = , determinamos el número número de divisiones del nonio, nonio, entonces:
=
10 min× 60 .1 min 10
⇒
10 × 60 10
= 60
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Planimetría_____________________________________________________________
Es aquel en el que el número de divisiones es una menos que el de divisiones del segmento correspondiente de la escala principal, por ejemplo: divisiones del vernier tendrán la longitud igual igual a divisiones divisiones de la escala; en consecuencia cada división del vernier será mas larga que una de la escala principal . .
En los trabajos de topografía se hace necesario fijar el transito en una posición cualquiera, para esto el equipo se encuentra dispuesto de unos sistemas de abrazadera, la cual ejerce una presión en la sección de cono donde gira el montante o el telescopio, impidiendo así estos movimientos. movimientos. El sistema se opera por medio de unos tornillos a los que denominamos radiales; ya que se encuentran dispuestos en el transito en posición radial con respecto a los limbos , sin embargo luego de fijar, ya sea la alidada o el telescopio, se le puede imprimir un movimiento pequeño a él conjunto conjunto (abrazadera (abrazadera y cono cono interior), moviendo un tornillo que denominamos tangencial, por su posición tangente con respecto al limbo del sistema que se está moviendo, por ejemplo si el sistema que este fijo es el del movimiento del telescopio, el tornillo de movimiento fino estará tangente tangente al limbo vertical. En este sistema sistema lo que hace el tornillo es que al
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Capitulo 3: Teodolito
Los equipos antiguos tenían un nivel tubular, que se utilizaba para asegurar que la línea de índice del círculo vertical quedara paralela al eje óptico cuando éste estuviera en horizontal, antes de cada medida. En la actualidad para no tener que realizar esta operación tan molesta y que demoraba los trabajos, se emplean unos dispositivos que reflejan la inclinación inclinación del eje vertical vertical para para compensarlo. compensarlo. Esto fue desarrollado por la empresa Askania de Berlín, después de la segunda guerra mundial. El primer sistema consistía en un péndulo compensador, en el que la lectura se efectúa a través de un arreglo óptico, con un prisma unido a un péndulo, que cuando el teodolito se encuentra inclinado con respecto a la vertical, el péndulo actúa en dirección opuesta a la de la inclinación; esta influencia nos ayuda a que la lectura que se obtiene en el índice no cambie debido a la inclinación del eje vertical y se correcta.
En general estos sistema de compensación tiene unos intervalos de acción grandes pero lo mejor es realizar una correcta nivelación de transito para así ayudar a todo el conjunto en su operación y evitar que en casos donde el compensador esté fallando se presenten errores por ésta causa, aunque no sean de una gran magnitud. Por ejemplo Cuando el sistema de nivelación solo se encuentra apoyado de un nivel de blanco, el cual tiene una precisión del del centrado de ± 3 minutos el
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Planimetría_____________________________________________________________
Para poder efectuar la lectura de los limbos que se encuentran dentro del transito, existen una serie de prismas y lentes que tiene el propósito de iluminar y llevar la imágenes de los limbos y los verniers a unos dispositivos que denominamos microscopios de lectura; lectura ; que son en los que se se ven las imágenes imágenes de los limbos limbos y los vernier en escalas mas grandes; de estos se encuentra de dos tipos: tipos: microscopio microscopio a escala y microscopio con nonio óptico (micrómetro). En los tránsitos los microscopios puede venir en el montante o a un lado del telescopio y de acuerdo con el sistema de medida del instrumento mostrar diferente
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Capitulo 3: Teodolito presenten este sistema son utilizados en trabajo que requieran poca requieran poca precisión debido a que en ellos una eventual excentricidad del limbo generaría errores en las lecturas.
En éste el sistema óptico de lectura permite ver simultáneamente en el microscopio dos lugares diametralmente opuestos del limbo (uno 180° más que el otro), lo que equivaldría a realizar una lectura con el microscopio a derecha del transito y una a izquierda se eliminan así los errores de excentricidad. Para efectuar una lectura en este sistema se opera de la siguiente forma: como ya se dijo en el microscopio se ven simultáneamente las dos partes del circulo ( una con numeración de derecha derecha a izquierda izquierda y la la otra con numeración de izquierda a derecha, por eso cuando giramos el transito se ve un desplazamiento en sentido
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Planimetría_____________________________________________________________
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Capitulo 3: Teodolito En los equipos de lectura por coincidencia existe un botón que se llama conmutador , con el que podemos cambiar cambiar la imagen en él microscopio de las dos partes del círculo horizontal horizontal por la imagen del círculo vertical y viceversa.
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Planimetría_____________________________________________________________
Es la parte del telescopio que se antepone al ojo del observador, que presenta una disposición disposición de lentes la imagen del objeto que se encuentra a la distancia. Un ejemplo de un ocular ocular la podemos apreciar en la Los oculares poseen las siguientes características:
• •
Aumento visual Campo visual
Es una placa de cristal sobre la cual se encuentran grabados un trazo vertical y otro horizontal que nos permiten realizar la puntería a los puntos que se visen, en esta placa es donde se forma la imagen generada por los rayos que atraviesan el objetivo. En algunos equipos la mitad inferior del la retícula presenta un doble trazo para facilitar y encuadrar con mayor exactitud el hilo de la plomada o las señales muy lejanas. En otros equipos se pueden encontrar unos trazos horizontales más pequeños pequeños (hilo superior, superior, hilo hilo inferior) inferior) que son empleados para medida estadimétricas de
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Capitulo 3: Teodolito con rosca interna y otro adentro del telescopio con rosca externa, por esto es que al imprimirle movimiento al cilindro cilindro expuesto el segundo segundo se mueve mueve con respecto a éste. éste.
Es un conjunto de lentes que se encuentra frente al objeto visado (espacio objeto) con la función de captar la luz procedente de éste y dirigirla al resto del sistema. Las características del objetivo son: • Luminosidad • Campo visual
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Planimetría_____________________________________________________________
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Capitulo 3: Teodolito
Este se presenta cuando la condición tres de la fig. 2 no es cumplida. El error de colimación horizontal es el error en dicho ángulo divido a la falta de perpendicularidad del eje ZZ y el eje HH. Este puede ser corregido por procedimientos de de campo y unos cálculos. Para determinar si se presenta el error de colimación se debe debe visar a un punto A que se encuentra por conveniencia conveniencia a una distancia mayor a 150m, 150m, con el telescopio en posición I registrando el el ángulo. Luego se lleva el equipo a posición II y se visa de nuevo nuevo al punto A. esta nueva lectura menos 180º 180º debe ser igual a la lectura inicial, teniendo en cuenta la precisión que puede ser exigida al instrumento. Si no se presenta de esta forma es porque el transito sufre de colimación horizontal. La diferencia entre el ángulo inicial y el segundo ángulo dividido en dos da la magnitud magnitud del error de colimación. Un procedimiento de campo para corregir la influencia del error de colimación es trabajar con visuales dobles (posición I y II) en las cuales se anotan los resultados obtenidos y la media de estos es igual al ángulo correcto. correcto.
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Planimetría_____________________________________________________________ y enfo enfoca carr hasta lograr que al desplazar la cabeza no se note ningún tipo de movimiento entre la entre los trazos del retículo y la imagen.
Esto es causado por imperfecciones imperfecciones en la graduación de los limbos en fábrica. fábrica. El efecto de dicho errores puede ser reducido empleando métodos de medición de ángulos como el de repeticiones y reiteraciones2. Es de gran gran importancia realizar la aclaración aclaración de que la precisión de medida angular no se aumenta aumenta solo solo con aumentar el el numero numero de n repeticiones repeticiones de una puntería, puntería, debido debido a que la condiciones atmosféricas, las las del observador y el instrumento también imponen limites, limites, de allí que para un numero n de observaciones que se ven afectadas afectadas por un Emc el error de la media media es igual:
=
Para comprender mejor esto es recomendable acercarse a la lectura del capitulo capitulo de errores. Si
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Capitulo 3: Teodolito
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Planimetría_____________________________________________________________
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Capitulo 3: Teodolito
Esta situación hace que la línea de colimación describa un plano inclinado al invertir el anteojo y, por tanto, si las visuales hacia atrás y hacia delante tienen ángulos diferentes de inclinación, originan ángulos horizontales con error. Los errores por esta causa pueden eliminarse promediando un número igual de lecturas directas e inversas.
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Planimetría_____________________________________________________________ Esta situación produce un error en todos los ángulos medidos desde un mismo punto cuya magnitud varía con la dirección y inversamente a la longitud de la misma. Debe verificarse a intervalos la posición de la plomada común o de la plomada óptica, para asegurarse que permanecerá centrada y que el instrumento esta
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Planimetría________________________________________________________________
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Capitulo 4: Ángulos
Los rumbos representan un sistema para designar las direccio nes de las líneas. El rumbo de
una línea es el ángulo ángulo horizontal agudo agudo medido entre la línea Norte-Sur Norte-Sur de referencia hacia el Este o el Oeste.
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